fr.cnes.sirius.patrius.math.optim.joptimizer.functions

Class QuadraticMultivariateRealFunction

java.lang.Object

fr.cnes.sirius.patrius.math.optim.joptimizer.functions.QuadraticMultivariateRealFunction

All Implemented Interfaces:

TwiceDifferentiableMultivariateRealFunction

Direct Known Subclasses:

LinearMultivariateRealFunction, PSDQuadraticMultivariateRealFunction

public class QuadraticMultivariateRealFunction
extends Object
implements TwiceDifferentiableMultivariateRealFunction

Represent a function in the form of
f(x) := 1/2 x.P.x + q.x + r
where x, q ∈ Rⁿ, P is a symmetric nXn matrix and r ∈ R.
NOTE1: remember the two following propositions hold:

1. A function f(x) is a quadratic form if and only if it can be written as f(x) = x.P.x for a symmetric matrix P (f can even be written as x.P1.x with P1 not symmetric, for example
   f = x^2 + 2 x y + y^2 we can written with P={{1, 1}, {1, 1}} symmetric or with P1={{1, -1}, {3, 1}} not symmetric, but here we are interested in symmetric matrices for they convexity properties).
2. Let f(x) = x.P.x be a quadratic form with associated symmetric matrix P, then we have:
   • f is convex <=> P is positive semidefinite
   • f is concave <=> P is negative semidefinite
   • f is strictly convex <=> P is positive definite
   • f is strictly concave <=> P is negative definite

NOTE2: precisely speaking, this class should have been named "PolynomialOfDegreeTwo", because by definition a quadratic form in the variables x1,x2,...,xn is a polynomial function where all terms in the functional expression have order two. A general polynomial function f(x) of degree two can be written as the sum of a quadratic form Q = x.P.x and a linear form L = q.x (plus a constant term r):
f(x) = Q + L + r
Because L is convex, f is convex if so is Q.

Since:

4.6

Author:

alberto trivellato (alberto.trivellato@gmail.com)

See Also:

"Eivind Eriksen, Quadratic Forms and Convexity"

Field Summary

Fields

Modifier and Type	Field and Description
protected RealMatrix	p Quadratic factor.

Constructor Summary

Constructors

Constructor and Description
QuadraticMultivariateRealFunction(double[][] pMatrix, double[] qVector, double rValue) Constructor
QuadraticMultivariateRealFunction(double[][] pMatrix, double[] qVector, double rValue, boolean checkSymmetry) Constructor

Method Summary

Modifier and Type	Method and Description
int	getDim() Get dimension
double[]	gradient(double[] xValue) Function gradient at point X
double[][]	hessian(double[] xValue) Function hessian at point X.
double	value(double[] xValue) Evaluation of the function at point X

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

p

protected final RealMatrix p

Quadratic factor.

Constructor Detail

QuadraticMultivariateRealFunction

public QuadraticMultivariateRealFunction(double[][] pMatrix,
                                         double[] qVector,
                                         double rValue,
                                         boolean checkSymmetry)

Constructor

Parameters:

pMatrix - matrix P

qVector - vector Q

rValue - value

checkSymmetry - check the symmetry?

QuadraticMultivariateRealFunction

public QuadraticMultivariateRealFunction(double[][] pMatrix,
                                         double[] qVector,
                                         double rValue)

Constructor

Parameters:

pMatrix - matrix P

qVector - vector Q

rValue - value

Method Detail

value

public final double value(double[] xValue)

Evaluation of the function at point X

Specified by:

value in interface TwiceDifferentiableMultivariateRealFunction

Parameters:

xValue - point

Returns:

evaluation

gradient

public final double[] gradient(double[] xValue)

Function gradient at point X

Specified by:

gradient in interface TwiceDifferentiableMultivariateRealFunction

Parameters:

xValue - point

Returns:

gradient

hessian

public final double[][] hessian(double[] xValue)

Function hessian at point X.

Specified by:

hessian in interface TwiceDifferentiableMultivariateRealFunction

Parameters:

xValue - point

Returns:

hessian

getDim

public int getDim()

Get dimension

Specified by:

getDim in interface TwiceDifferentiableMultivariateRealFunction

Returns:

dimension